Suppose \(X\)  is a random variable defined on \((\Omega , \mathcal {F}, \mathbb {P}_{\Omega })\)  which has a non-zero mean and admits a moment generating function. Let \(X_1, X_2, \ldots , X_n\sim X\)  be i.i.d.  random variables. Denote \(V_n(s)=W_n(s)/(\mu \sqrt{n})\)  for brevity. Minkowski's inequality [1]} and the fact that \(X_1, X_2, \ldots , X_n\sim X\)  are identically distributed imply
\(\Big (\mathbb {E}_{\Omega }|V_n(s)|^j\Big )^{1/j}&=\mu ^{-1}n^{-3/2}\Big (\mathbb {E}_{\Omega }|\sigma _1X_1+\sigma _2X_2+\cdots +\sigma _nX_n|^j\Big )^{1/j}\\&=\mu ^{-1}n^{-3/2} \Vert   \sigma _1X_1+\sigma _2X_2+\cdots +\sigma _nX_n \Vert _j\\&\le \mu ^{-1}n^{-3/2}\Vert  X \Vert _j \sum _{k=1}^n \sigma _k.\) 
